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5 COMPARISON OF SOLUTIONS OF DIFFERENTIAL EQUATIONS 285
to Fr, for it is analytic in Jr since s ->f(s9 y(sj) is (9.7.3); and its continuity in
Jr at once follows from (8.11.1). We therefore have defined a mapping g of Vr into Fr, and the end of the proof is then unchanged. |
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(10.4.6) Remark. The proof of (10.4.5) shows that the result is still valid
when K = R and when / satisfies the following weaker hypotheses: (a) for every continuous mapping / -> w(t) of I into H, t -*/(/, w(0) is a regulated function in I (Section 7.6); (b) for any point (t, x) e I x H, there is a ball J of center / in I and a ball B of center x in H such that/is bounded in J x B, and there exists a constant k ^ 0 (depending on J and B) such that \\f(s,yi)-f(s,y2)\\ <£|bi -y2\\ for ^eJ, yi9y2 in B. Such a function/ is said to be locally lipschitzian in I x H; equation (10.4.2) is then to be understood as holding only in the complement of an at most denumerable subset of J. This last remark also enables one to replace the open intervals I and J by any kind of interval in R. |
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5. COMPARISON OF SOLUTIONS OF DIFFERENTIAL EQUATIONS
We say that a differentiate mapping u of an open ball J c I into H is an
approximate solution of (1 0.4.1 ) with approximation & if we have |
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for any t e J.
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(10.5.1) Suppose || D2 /(/,*) || < k In I x H. If u, v are two approximate
solutions of (1 0.4.1 ) in an open ball J of center t0 , with approximations S1,s2> then, for any t e J, we have
(10.5.1.1) KO - KOII < Wo) - *>(*o)ll e*""'0' + fa + B2) ^ *' " 1 .
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(For k = 0, (eklt~*°\ - l)/k is to be replaced by \t - r0|). We immediately are
reduced to the case K = R, tQ = 0 and t ^ 0 by putting t = t0 4- ag, \a\ = 1, ^ > 0; then ifu^) = u(t0 + a£)9 ^(£) = v(t0 + a^), wx and v^ are approximate solutions of jc' = q/"(/0 + a^ ^™1^)> whence our assertion. From the relation \\u'(s) —f(s9 u(s))\\ < 8! in the interval 0 < ,y < /, we deduce by (8.7.7) |
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- | f(s9u(s»ds
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